dripplons as localized and superfast ripples of water confined …lbocquet/publi/163article.pdf ·...

ARTICLE

Dripplons as localized and superfast ripples ofwater confined between graphene sheetsHiroaki Yoshida 1,2, Vojtěch Kaiser 1, Benjamin Rotenberg 3 & Lydéric Bocquet1

Carbon materials have unveiled outstanding properties as membranes for water transport,

both in 1D carbon nanotube and between 2D graphene layers. In the ultimate confinement,

water properties however strongly deviate from the continuum, showing exotic properties

with numerous counterparts in fields ranging from nanotribology to biology. Here, by means

of molecular dynamics, we show a self-organized inhomogeneous structure of water confined

between graphene sheets, whereby the very strong localization of water defeats the energy

cost for bending the graphene sheets. This leads to a two-dimensional water droplet

accompanied by localized graphene ripples, which we call “dripplon.” Additional osmotic

effects originating in dissolved impurities are shown to further stabilize the dripplon. Our

analysis also reveals a counterintuitive superfast dynamics of the dripplons, comparable to

that of individual water molecules. They move like a (nano-) ruck in a rug, with water

molecules and carbon atoms exchanging rapidly across the dripplon interface.

DOI: 10.1038/s41467-018-03829-1 OPEN

1 Laboratoire de Physique Statistique, Ecole Normale Supérieure, UMR CNRS 8550, PSL Research University, 24 rue Lhomond, 75005 Paris, France. 2 ToyotaCentral R&D Labs., Inc., Nagakute, Aichi 480-1192, Japan. 3 Sorbonne Université, CNRS, Physicochimie des électrolytes et nanosystèmes interfaciaux, UMRPHENIX, F-75005 Paris, France. Correspondence and requests for materials should be addressed to L.B. (email: [email protected])

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http://orcid.org/0000-0002-7283-8617http://orcid.org/0000-0002-7283-8617http://orcid.org/0000-0002-7283-8617http://orcid.org/0000-0002-7283-8617http://orcid.org/0000-0002-7283-8617http://orcid.org/0000-0002-6129-4618http://orcid.org/0000-0002-6129-4618http://orcid.org/0000-0002-6129-4618http://orcid.org/0000-0002-6129-4618http://orcid.org/0000-0002-6129-4618http://orcid.org/0000-0001-5198-4650http://orcid.org/0000-0001-5198-4650http://orcid.org/0000-0001-5198-4650http://orcid.org/0000-0001-5198-4650http://orcid.org/0000-0001-5198-4650mailto:[email protected]/naturecommunicationswww.nature.com/naturecommunications

Nano-confined water appears in a wide variety of fieldssuch as biology, geology, or tribology, and has longattracted significant attentions1–3. The recent emergenceof two-dimensional materials such as graphene and grapheneoxide (GO) membranes now offers the possibility to explorefundamentally the properties of water down to the molecularscale4–13, together with direct applications in the fields of energyconversion, water desalination, dehydration or proton conductionin fuel cells10,14–20. Water confined in such 2D materials behavesquite differently from bulk water, leading e.g., to highly orderedstructures even at room temperature, with lattice structures neverobserved in bulk ice, as revealed by means of both simulationsand experiments21–26. Furthermore, peculiar dynamical proper-ties have been observed for water flowing through nanoporousmembranes, such as a fast transport and significant selectivity ofsolutions27–30. Understanding the behavior of confined water isthus crucial in order to control and optimize the performance ofsuch new materials.

Water between two parallel solid plates is known to formordered phases such as mono- and bi-layers, depending on theinterlayer distance between the solid plates, as evidenced fromvarious experiments e.g., in clays31–34, between mica and glass35,mica and graphene36, graphene layers23, and in GO37. In thegraphene cases, elasto-capillary couplings of water phases withthe confining substrate are not obviously expected owing to thehigh bending rigidity of the surfaces, graphene having a highbending modulus in the range of B ~ eV ≈ 40kBT at room tem-perature. One may still define an elasto-capillary length as‘ ¼ ffiffiffiffiffiffiffiffiffiffiffiB=Δγp , by comparing bending to a typical surface energyΔγ38. One obtains ‘ � 1 nm (with Δγ ≈ 0.07 Jm−2 to fix orders ofmagnitude). This suggests indeed that water confined at sub-nanometer scales may actually couple to the graphene elasticityvia surface energy.

In the present study, we investigate a peculiar inhomogeneitycaused by the coupling mentioned above in a two-dimensionalwater thin film confined between graphene sheets, revealed usingmolecular dynamics (MD) simulations. A self-organized structureof water arises from the balance between the layering tendency ofwater and the stiffness of the graphene sheets. An example isshown in Fig. 1 (see also Supplementary Movies 1 and 2). Spe-cifically, when the density of water molecules contained betweenthe graphene sheets exceeds a threshold, a two-dimensionaldroplet of localized water is formed by bending the graphenesheets—here we call this droplet “dripplon.” It has recently beenshown that an isolated water droplet maintains its localizedstructures between two graphene sheets against the elasticity39.Here, we demonstrate the localization of water in water caused bythe disjoining contributions, which is further relevant for lamellar

GO-like structures. We propose below a detailed understandingfor the creation of this structure, coupling calculation of the freeenergy and disjoining pressure of mono- and bi-layers of water,together with a thermodynamic elasto-capillary modelization.The fast dynamics of the dripplon will be finally highlighted.Overall, the detailed implementation of the MD simulations isdescribed in Supplementary Note 1. We quote that several waterand graphene models have been tested, allowing to assess therobustness of our results.

ResultsWater confined between rigid graphene sheets. We start byconsidering the behavior of water between two rigid graphenesheets. In these first MD simulations, the graphene sheets areplanar and constrained to move only in the perpendiculardirection under a fixed pressure of 1 atm, whereas the fluid ismaintained at a temperature of 300 K. The lateral size of thesheets shown in Fig. 2 is S= 3.9 × 4.3 nm2, but the behavior isessentially the same for larger sheets within the range we checked,up to S= 15.7 × 17.0 nm2 (see Supplementary Note 1.) In Fig. 2a,the interlayer distance h between the two rigid graphene sheets insteady state is plotted as a function of the surface number densityρ of water molecules (number of water molecules per unit area).Although h increases almost linearly for high surface density, aplateau of h ≈ 0.95 nm is observed in the range ρ < 24 nm−2. Inthe latter case an ordered structure with a water bi-layer isformed. For ρ≲ 13 nm−2, another plateau with h ≈ 0.67 nm cor-responds to a water monolayer. In this range, both the mono- andbi-layer are observed depending on the initial configuration,suggesting that both states are metastable under these conditions.The bi-layer at low density is associated with the nucleation of aliquid-vapor interface, the density of the liquid-like regionremaining almost unchanged. Finally, a third state (h ≈ 0.8 nm) isalso observed in the narrow range of 13 nm−2 < ρ < 14 nm−2.Apart from their coexistence, water mono- and bi-layers formedbetween solid plates have widely been studied experimentally, e.g.,in the crystalline swelling regime of clays31–34 or the direct forcemeasurement between surfaces40,41. The step ~ 0.28 nm observedhere for the thickness of the water layer is in agreement withthose experimental results.

The in-plane structure of the water mono and bi-layers isgenerally liquid-like, as observed e.g., in clays. Only for themonolayer with the highest density (at ρ= 13.1 nm−2, on thelower plateau in Fig. 2a), close to the onset of coexistence, do weobserve a square lattice with lattice parameter ~ 2.8 Å resemblingthe square-ice-like structure reported in ref.23. However, such along-range order disappears if the confining graphene sheets are

Entire system

0

10

15

5

(nm

)

20

10 1550

10

9

7

8: Water: Carbon

Top view Gap distribution

Gap

(nm)

(Å)

6.4

Fig. 1 Simulated system, and a schematic representation of the two-dimensional droplet “dripplon”. The two flexible graphene sheets with water moleculesin between are sandwiched by two water reservoirs (fixing a hydrostatic pressure of 1 atm). The top view of the confined water molecules is also shown,along with the corresponding figure showing the distribution of the gap between the two graphene sheets. The water molecules in the reservoirs are lightcolored to focus on the confined water. The water molecules belonging to the dripplon are color-highlighted in red, whereas the confined water moleculesoutside the dripplon are blue-colored

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flexible, as considered in the following sections, and only the localordering may survive.

To confirm the coexistence of multiple stable states at lowdensity, we show the free energy landscape in Fig. 2b. The freeenergy per unit area G is plotted as a function of h, for the fivevalues of ρ indicated by the vertical lines in Fig. 2a. For eachdensity, we performed series of MD simulations with fixedinterlayer distances h during which the pressure p(h) wascomputed. The (swelling) free energy is then obtained usingGðhÞ ¼ p0h�

R hh0pðh′Þdh′. Here, p0= 1 atm and h0 is arbitrarily

chosen such that G= 0 at the local minimum at h > 0.9 nm.Overall, the diagram in Fig. 2a is consistently explained by thefree energy landscape. Specifically, there are two local minima forρ= 9.5 nm−2 corresponding to the mono- and bi-layer states,whereas only one local minimum at larger distance is found for ρ= 23.9 nm−2. For ρ= 13.4 nm−2 and 13.7 nm−2, a shallow wellcorresponds to the state with h ≈ 0.8 nm.

The nucleation of a liquid-vapor interface under someconditions, despite the entailed energetic cost, points to the highcohesive energy of water. The strong localization tendency ofwater results from its interaction with itself rather than with theconfining walls and is also at the origin of the hydrophobicbehavior of surfaces42. In the present case of extreme confinementdown to the molecular scale, the instability of intermediatehydration states is also owing to the packing of discrete watermolecules. The overall driving force for phase separation betweenmixed hydration states can be estimated in our molecular

simulations by considering systems too small to accommodatetwo phases separated by an interface, which therefore remainhomogeneous. The interlayer distance obtained for a system of S= 1.2 × 1.3 nm2 is shown in Fig. 2a (red-dashed line). A singlehydration state is observed, whatever the initial value of theinterlayer distance. More quantitatively, the free energy G for twodensities is measured, to compute the disjoining pressure Π=−∂G/∂h, as plotted in Fig. 2c. The states are always unstable forthe negative disjoining pressure (Π < 0). Furthermore, in therange where ∂Π/∂h > 0, spinodal decomposition is expected oncethe homogeneity constraint is released. As we now proceed toshow, the response of the system to this driving force for phaseseparation also depends on the flexibility of the graphene sheets,which results in a peculiar inhomogeneous interlayer structure.

Graphene flexibility and localized dripplons. We now relax theconstraint of rigid graphene sheets and investigate how the water-graphene structure evolves. As depicted in Fig. 1, the graphenesheets are sandwiched by water reservoirs, which maintain ahydrostatic pressure of 1 atm (using pistons consisting of rigidgraphene sheets at the boundaries of the reservoirs). For thepurpose of examining the evolution of the hypothetical homo-geneous state, let us consider a situation with an initial homo-geneous state at h= 0.85 nm, which is the equilibrium state for anaverage number density ρav= 14 nm−2 (Fig. 2c). If the densitydrops to ρav= 12.1 nm−2, the system becomes unstable and the

a

c

Inte

r-la

yer

dist

ance

h (

nm)

Sideh

0

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0 10 20 30 40 50

: � = 12.1 nm–2

: � = 14 nm–2

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Number density � (nm–2)

Top

Dis

join

ing

pres

sure

� (×

103

atm

)

Slope = 5.7×103 atm/nm

Inter-layer distance h (nm)

b

: � = 9.5 nm–2

: 13.1 nm–2

: 13.7 nm–2

: 14.3 nm–2

: 23.9 nm–2

–40

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0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

Dis

join

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�

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

d

Fre

e en

ergy

G (

kJ m

ol–1

nm

–2)


� = 12.1 nm–2

� = 14 nm–2


Fig. 2 Properties of water confined between two rigid graphene sheets. a Interlayer distance h versus number density per unit area ρ, obtained for thesystem of water molecules between two rigid graphene sheets of size 3.9 × 4.3 nm2 under a pressure of 1 atm. The top views of the water configuration andthe side views of the water and rigid graphene sheets at several points are presented inside the graph. The red-dashed line shows the result obtained forthe small size system (1.2 × 1.3 nm2.) b Free energy per unit area G at several values of ρ, along the vertical lines shown in a. The reference energy for eachvalue of ρ is chosen such that G= 0 at the local minimum in h > 0.9 nm. c Disjoining pressure Π vs interlayer distance h for the small size system (1.2 × 1.3nm2), at ρ= 12.1 and 14 nm−2. The red line shows the slope of 5.7 × 103 atm nm−1 used in the analysis of spinodal decomposition process as schematicallyillustrated in d

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phase separation takes place by spinodal decomposition, as illu-strated in terms of the disjoining pressure in Fig. 2d. Althoughsuch a sudden decrease in the density is artificial, it allows toquantitatively investigate this instability which may occur duringthe drying process of e.g., a GO membrane.

Figure 3a shows snapshots illustrating the evolution of theinitially homogeneous state at h= 0.85 nm, together with thedistributions of the gap between carbon atoms of the upper andlower graphene sheets. (See Supplementary Movie 2 for thecorresponding video). Here we clearly observe the instability andthe phase separation into the monolayer water phase with h ≈0.67 nm and the double-layer water phase with h ≈ 0.95 nm. Theresulting structure consists of a two-dimensional water dropletbetween the flexible graphene sheets, as illustrated in Fig. 1 (seealso Supplementary Movie 1). Forming such a two-dimensionaldroplet requires overcoming the cost of the elastic energy tocreate the ripples in the graphene sheets and that of forming theinterface between water mono- and bi-layers. We call theresulting combination of droplet and ripples a “dripplon.”

The initial stage of the instability (Fig. 3a at t= 5 ps),understood as the spinodal decomposition discussed above, canbe discussed more quantitatively. In Fig. 3b, we plot the scatteringintensity as a function of the wave number, computed using thebinary image of the interlayer distance distribution at t= 5 ps, asshown in the inset (we used five independent snapshots for betterstatistics, see Supplementary Note 2). The spectrum shows a log-normal distribution because the wave number has a lower bound

due to the size of the simulation box43. The fastest growing modeof the decomposition process is then obtained as λm= 2π/qm=5.7 nm, where qm is the wave number at the peak. We checkedthat this value was insensitive to the size of the simulation box.

This characteristic length for the spinodal structure can beestimated by extending the classical theory for the spinodaldecomposition of thin films44,45. In the present case, theinterfacial cost associated with the existence of a dripplonoriginates mainly from the elastic deformation of the graphenebetween the monolayer and bi-layer regions. Accordingly thesurface energy cost is merely replaced by the elastic energy tobend the graphene sheets. Assuming a slight deformation of thefilm thickness, say δh(x, y), the total free energy cost writes

G½δh� ¼Z

dS12BðΔδhÞ2 þ 1

2∂2G∂h2

� �0

δh2� �

; ð1Þ

where B is the bending modulus of graphene, Δ stands for the

Laplacian and ∂2G∂h2

h i0≡− ∂∂h

� �0 (with Π the disjoining pressure

and G the free energy per unit surface introduced above). Now,for a negative compressibility, i.e., [∂Π/∂h]0 > 0, a prototypicalspinodal scenario occurs and the fastest growing mode λm is

a

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(nm) (nm)

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: dripplons

: dripplons with solute particles

0

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d

I (q)

(a.

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R/R

0

t1/3 (ps1/3) (nm)

Fig. 3 Spinodal decomposition and osmotic stabilization of dripplons. a Time evolution of the top views of the water molecules between flexible graphenesheets, starting from the homogeneous state at h= 0.85 nm. Each panel is accompanied by the distributions of gap between upper and lower flexiblegraphene sheets. See also Supplementary Movies 1 and 2. b Radially averaged scattering intensity function of the binary image (inset) of dripplons at t= 5ps; five images from independent simulations are used for better statistics. The red curve shows a fit with a log-normal distribution. c Time evolution ofinitially two identical dripplons; the radius R, normalized by the initial radius R0, is plotted as a function of t1/3. The corresponding gap distributions ofinitially two identical dripplons are shown in d. See Supplementary Movie 3 for the video




expressed as λm= 2πξ, with

ξ ¼ B∂Π=∂h½ �0

� �1=4: ð2Þ

Numerically, using the values for the disjoining pressureprovided in Fig. 2c, yielding [∂Π/∂h]0 ≈ 5.7 × 103 atm nm−1, weobtain the value of ξ= 0.8 nm for the fastest growing mode. Thisvalue is in excellent agreement with the value measured in thesimulation images, yielding ξmeasured= λm/2π= 0.9 nm. Thisdemonstrates that the driving force associated with water layering(disjoining pressure) is able to balance the considerable cost ofinterfacial energies originating in graphene deformation. This is akey for the dripplon existence.

Ostwald ripening and dripplon osmotic stabilization. After theinitial decomposition process, the number of dripplons gra-dually decreases, until a single dripplon subsists (Fig. 3a, t= 30~ 400 ps). In order to examine this long-time behavior ofmultiple dripplons, we show the evolution of dripplon radii inFig. 3c starting from a state having two identical dripplons inthe same plane (prepared by replicating a smaller system with asingle dripplon). The corresponding gap distributions are alsoshown in Fig. 3d (and in Supplementary Movie 3.) One of thedripplons shrinks over time, whereas the other grows until asingle dripplon survives, without direct collisions. This processis interpreted as a two-dimensional version of Ostwald ripen-ing46,47. The theory of Ostwald ripening predicts that theaverage radius of the drop evolves as R(t) ~ t1/3. The plot of theradius as a function of t1/3 does display a linear dependency,apart from the very final stage of vanishing the smaller drip-plon. This implies that the process is indeed driven by theminimization of the interfacial energy, which arises at theperimeter of the dripplon.

However this ripening can be counteracted and dripplonscan be stabilized at equilibrium. It is indeed well known thatdispersed (nano-)droplets can be stabilized against Ostwaldripening process if they contain trapped species that areinsoluble in the major phase48,49. Similarly here, adding a solutethat is only stable in the water double-layer is accordinglyexpected to stabilize the dripplon. An osmotic pressure buildsbetween the inner dripplon (with double water layers) and theouter region (with a single water layer), and it compensates at

equilibrium the interfacial pressure associated with thedripplon interface between the mono- to the bi-layer of water.The latter is defined in terms of a two-dimensional line tensionγ, which accounts for the elastic deformation of the graphenesheets at this dripplon interface. A rough estimate of thecorresponding elastic energy suggests that

γ � B‘; ð3Þ

with ‘ a molecular length. This yields typically γ ≈ 3 ~ 4 × 10−10

J m−1 (with B= 2 eV and ‘ � 1 nm). The pressure balance thusleads to an estimate of the stable radius as

γ

Rs� ρskBT; ð4Þ

where ρs is the solute surface concentration (see SupplementaryNote 3 for details). Using ρs= xsρ2 with ρ2 ’ 20 nm−2 thedensity inside the dripplon and xs the molar fraction of solute—say, xs ~ 10% to fix order of magnitudes—, one obtains anequilibrium radius of the dripplon of Rs ≈ 40 nm. Such anequilibrium dripplon size is completely relevant experimentallybut is currently out of reach for molecular simulations.

In the MD simulations, we examined however the effect ofadded solute particles in dripplons, using a feasible system size.As shown in Fig. 3c, we indeed observe as expected a slowdown ofthe coarsening ripening process. Here, we put three electricallyneutral particles in each dripplon in Fig. 3c (see SupplementaryMovie 3 for the corresponding video); the solutes are modeledwith a Lennard-Jones interaction potential with a diametercomparable to two water molecules, so that they are less solublein the water monolayer than in the bi-layer (see SupplementaryNote 3). Such a finding not only confirms the mechanism at playfor the formation of dripplons, but also suggests how suchdripplon structures are expected to be stabilized by impurities inexperiments.

Thermodynamics of dripplon creation. We now come back onthe creation of the dripplon structure. As shown in Fig. 4a, theMD results demonstrate that there is a minimum water densityfor the creation of the dripplon. We thus explore further thethermodynamics of the dripplon and propose a model for thisthreshold mechanism.

: test particle method

: �0 = –27.3 kJ/mol

: quadratic fit

–800

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5 10 15 20 25 30

2R

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ius

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Fre

e en

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G (

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ol–1

nm

–2)

Number density � (nm–2)

a b

Fig. 4 Thermodynamic model of dripplon creation. a Radius of the dripplon R versus average number density ρav, for the system size of 15.7 × 17.0 nm2. Thesolid line indicates the prediction of the thermodynamic model in Eq. (5). The prediction of the lever rule is shown by the dashed line. b Free energy per unitsurface as a function of number density ρ. The symbols indicate the results obtained using the test particle insertion method. The dashed and dash-dottedcurves are quadratic fits used in the thermodynamic model for the dripplon creation. The slope of the straight lines, ∂G/∂ρ= μ0=−27.3 kJ mol−1,corresponds to the chemical potential of bulk water at 300 K and 1 atm




Let us consider a dripplon of radius R. The free energy costassociated with this structure can be written as

Gtotal ¼ S� πR2

G1 þ πR2G2 þ 2πRγ; ð5Þ

where S is the total area, and G1(ρ1) and G2(ρ2) are the freeenergies per area of the mono- and bi-layer phases, respectively.The last term is the free energy cost associated with the elasticbending of the graphene sheets at the dripplon perimeter. This isaccounted for by a line tension γ, which relates to the bendingmodulus as in Eq. (4). We note here that a relevant theoreticalapproach using energy minimization has also been taken toexplain the formation of bubbles on the surface of two-dimensional materials50, although there are essential differencesin main contributions to the total energy.

The free energy terms G1(ρ1) and G2(ρ2) are expected to exhibitminima associated with the mono- and bi- layer densities. This isevidenced in Fig. 4b, where we report the free energy calculatedfrom the MD simulations. In practice, we have used Widom’s testparticle insertion method to compute the chemical potentialμ(ρ) with the rigid graphene system51,52, see SupplementaryNote 4 for details of the test particle insertion method. The freeenergy is then obtained by thermodynamic integrationGðρÞ ¼ R ρρ0 μðρ′Þdρ′. As shown in Fig. 4b, two local minima areclearly observed at ρ1= 10.9 nm−2 and ρ2= 21.5 nm−2, corre-sponding to water mono- and bi-layer states. The slopes of thelinearly decreasing parts agree well with the reference chemicalpotential μ0=−27.3 kJ mol−1, which we obtained using the testparticle simulation for a bulk water at 300 K and 1 atm, also inagreement with the literature53. These results confirm that the

c d

a

Snapshot

Gap distribution

6

8

10

12

0 1 2 3

: R = 2.9 nm: 3.4 nm: 4.1 nm

: ∝t –0.75

103

: Brownian disk

: ∝t –0.5

: dripplon : dripplon (relative)

� =1.30

1

2

3

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50(nm)

t = 0 t = 1 ns t = 2 ns

10 1550 10 1550

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(Å)

b

(nm

)

(nm) (nm)

MS

D 〈Δ

r 2 〉

(nm

2 )

100

10–1

10–2

10–3100

Time t (ps)101 102

� =1

Diff

usio

n co

ef. D

(×10

–5 c

m2 /

s)

Radius R (nm)

Sur

viva

l pro

babi

lity

Sc

101

100

10–1

10–2

10–3

10–4

Time t (ps)

100 101 102

Gap

(Å

)

Time t (ns)

Fig. 5 Dripplon dynamics. a Displacement of a dripplon over time, compared with diffusion of water molecules initially in the dripplon, which is visualized inb. The water molecules belonging to the dripplon at t= 0 are colored in red in b. The average density is ρav= 12.3 nm−2, in the graphene sheets ofS= 15.7 × 17.0 nm2. See Supplementary Movie 4 for the corresponding video. c Mean-square displacement (MSD) of a dripplon as a function of time, for adripplon with R= 2.9 nm. The MSD highlights superdiffusive behavior, Δr2

� �ðtÞ � tα with α≈ 1.3. The inset reports the calculated apparent diffusioncoefficient of a dripplon (as defined in the main text). The solid line is a one-parameter fit as cR−2. d Survival probability within the dripplon. Sc(t) indicatesthe probability that a carbon atom participates in the dripplon for longer than time period t (larger than 1 ps). The gap histories measured for three carbonatoms are shown in the inset




free energies for the mono- and bi-layer states, G1(ρ1) and G2(ρ2),can be properly approximated by quadratic functions of thedensity: Gi(ρi)= Ci(ρi− θi)2+ bρi+ d. The parameters Ci, θi, b,and d are estimated from the MD free energy plot in Fig. 4b (seeSupplementary Note 5).

Now using this expression for the free energy, we can calculatethe equilibrium dripplon size R for a given value of ρav byminimizing Gtotal(ρ1, ρ2, R) in Eq. (5), under the constraint(S− πR2)ρ1+ πR2ρ2= ρavS (by means of the method of Lagrangemultipliers). Figure 4a compares the theoretical prediction for theequilibrium dripplon radius as a function of average density withMD simulations results. A very good agreement is obtained,yielding a value γ= 6.1 × 10−10 J m−1 for the line tension,adjusted to obtain a best fit. We quote that this value for γ is ingood agreement with the simple estimate suggested above,γ � B=‘.

Overall, the thermodynamic model agrees very well with thesimulation results. It predicts a minimal density for the creationof the dripplon, with the expression ρc ¼ θ1 þ ð3=2Þπγ2= 2C21S θ2 � θ1ð Þ

� �1=3. Using the parameters from the free

energy, one predicts the value ρc= 11.5 nm−2 for Fig. 4a, in verygood agreement with the MD data. The prediction of the “leverrule”, corresponding to γ= 0, fails to predict the observedthreshold density. As a last comment, no upper bound onthe dripplon size is expected. However on the simulation side, thesize of the numerical box should exceed the expected dripplonsize in order to avoid finite-size effects, see SupplementaryNote 5.

Superdiffusive dripplon dynamics. The dynamics of the drip-plon also highlights interesting features. In Fig. 5a, b (also inSupplementary Movie 4), we follow the motion of a dripplon andthe underlying dynamics of water molecules belonging to thedripplon. A first striking remark is that the dripplons are fastmoving objects, in spite of their extended nature (a dripplon withradius R= 3 nm contains typically 500 water molecules). This ishighlighted in Fig. 5c where we report the mean-squared dis-placement (MSD) of a given dripplon versus time, as well as theextracted apparent diffusion coefficient. First the long-timebehavior of the MSD is shown to be superdiffusive54, withΔr2h i � Γtα. The exponent is typically α ≈ 1.3, substantially largerthan unity. In the present case, one possible explanation for thissuperdiffusive behavior is the 2D nature of the dripplondynamics, which is expected to enhance logarithmically the dif-fusion coefficient (DðtÞ � log t at long time) via the long-timerelaxation of the hydrodynamic modes in 2D. However, it isinteresting to note that the superdiffusive behavior of the exten-ded dripplon contrasts with the bare diffusion behavior of indi-vidual water molecules, Δr2h iH2O� t, in spite of the 2D confininggeometry (see Supplementary Note 6).

Even more striking is that the superdiffusive dynamics of thedripplon as an extended object is nearly as fast as individual watermolecules, as highlighted by comparing the MSD, see Supple-mentary Note 6. If one defines an apparent diffusion coefficient ofa dripplon, as Dapp ¼ Γtα�10 =4 with t0= 0.2 ns to fix ideas55, thenthe values of Dapp are in the range of 10−5 cm2 s−1. (We checkedthat the velocity autocorrelation function decays in the time scaleof tens of ps, hereby supporting the choice of t0.) These valuescompare with the self-diffusion coefficient of the bulk water(≈2 × 10−5 cm2 s−1)56. This also compares with the diffusion ofwater molecules involved in dripplons (D= 2.4 × 10−5 cm2 s−1for water inside a dripplon—bi-layer—and 4.2 × 10−5 cm2 s−1 forwater outside—monolayer–, in agreement with literature forconfined water57). The motion of a dripplon with D ~ 10−5 cm2 s−1

thus appears as surprisingly fast for a collective object constituted

of hundreds of molecules. In Fig. 5c-(inset), we also show theapparent diffusion coefficient of dripplons (defined above) versustheir size. We find a decrease of the diffusion coefficient versusthe dripplon radius, scaling typically as ~ R−2, i.e., the inversedripplon area. This is reminiscent of a similar observation for themotion of a droplet of water on free-standing rippled graphene58,possibly suggesting that the energy dissipation is dominated bysurface friction.

To get further insights into the coupled water-dripplondynamics, we explored the relative motion of carbon sheets withrespect to the water. First in Fig. 5b, we mark water moleculesinitially inside a dripplon in the snapshots and compare theirdiffusive dynamics to that of the dripplon shown in Fig. 5a. Whatemerges from this graph is that water molecules do not follow thedripplon, but are actively exchanged across the dripplon interface.In other words, the dripplon, defined as a mono- to bi-layerinterface, constantly destroys and reforms itself while moving,just like a (nano-) “ruck in a rug”59: the motion of the dripplonstructure is thus by nature collective.

Further support of this picture emerges from the survivalprobability Sc(t) that a carbon atom belongs to the dripplon formore than a given time t, (out of those participating more than1 ps). As shown in Fig. 5d, the probability exhibits a clear decay,confirming the local creation/destruction mechanism for everyparticipating carbon atom. We compare the observed powerdecay, typically ~ t−β with β ≈ 0.75, with the correspondingprobability for a simplified model: namely the probability that afixed point on a plane stays within a disk moving in the 2D planeunder Brownian motion. This is a classical first-return problem,which leads to a survival probability decaying as ~ t−1/2, as can beconfirmed by means of a Brownian dynamics simulation of a disk(see Fig. 5d and Supplementary Note 6). The exponent β ≈ 0.75 isslightly larger than the Brownian prediction, which again can beattributed to the superdiffusive nature of the dripplon asdiscussed above. Additional to the carbon motion, we alsoexplored the water fluxes across the dripplon structure. To thisend, we computed the probability Pw(t) that a water moleculebelongs to the dripplon for a time period t (out of those inside thedripplon for more than 1 ps) and the corresponding survivalprobability SwðtÞ ¼ 1�

R t0PwðsÞds. This estimate indicates that

~97% of water molecules inside a dripplon pass across theinterface at least once within 100 ps, for all the dripplons up to R= 4.6 nm. This confirms a very active exchange of watermolecules across the dripplon interface, whereas keeping thedripplon shape.

Altogether, these observations suggest that the dynamics of thedripplon on longer time scales, including its diffusion, reflects acollective behavior which goes beyond that of the constitutingwater molecules and carbon atoms. The dripplon is permanentlydestroyed and reformed, leading to fast motion.

DiscussionWe have shown that the mixed hydration states of confined watermay couple to confining graphene flexibility to create localizedwater drops accompanied by a strong deformation of graphenesheets. The elastic bending energy cost creating the ripples isdefeated by the strong preference of water layering to homo-genization. The emergence of this localized structure is rationa-lized by means of a thermodynamic model, highlighting thecompetition between the disjoining pressure and surface elasticcosts of the graphene sheets. This points to an interesting analogybetween the behavior of molecularly confined water betweengraphene sheets, and that of thin liquid soap films. This is incontrast to the rigid layered materials with intercalating water,such as the swelling clays33, which are too rigid to sustain such




localized defects thus rather favoring the interstratification of theconfined water with different hydration states.

From an experimental point of view, such structures couldmanifest themselves in systems of layered graphene, such as withGO membranes. The condition for the creation of the dripplonrequires to reach the threshold density ρc, which is expected tooccur in the drying process of GO membranes. Also the typicallateral size of the sheets of the GO membrane is far larger (~ cm)than those considered in the present study29,30 and hence thewater density inside could be maintained in a quasi-stationarystate under ambient conditions, favoring the apparition of drip-plons. Let us also note that the molecular layering in the vicinityof the liquid–solid interface is quite general60, and it is experi-mentally observed for various liquids, such as alcohols andhydrocarbons61,62. The physical picture of the dripplon cantherefore be possibly applied to various liquids other than water,confined between graphene-based membranes.

Furthermore, we showed that solute particles and impuritiescontained in the dripplon lead to an osmotic stabilization of adripplon, against the Ostwald ripening. This suggests therobustness of equilibrium dripplons in real systems. Reversely,this solute-induced stabilization leads to a strong ‘wetting’ of thesolute particles within the dripplon: solutes and impurities aredressed in multi-layers of water and concentrated in these loca-lized structure. This effect could have an important role indetermining the structure of multilayer graphene-like GO whendried.

An interesting perspective concerns particle transport medi-ated by the dripplon dynamics. These fast moving ripples couldindeed act, to some extent, as cargoes transporting confinedparticles between the sheets. Indeed, the fast dynamics of theseextended objects could allow to boost the transport of particlesacross these layered materials by adding new degrees of freedomfor the strongly confined solute, thereby enhancing its dynamics.This is somewhat reminiscent of the coupling of water dynamicsto that of the (fluctuating) confining surfaces as shown recently inthe context of carbon nanotubes63. This suggests to explore theinterplay between the dynamics of the (fast) dripplon and that of(slow) suspended particles. From a more general perspective,these fast dripplon defects should impact water and ionic trans-port across such layered materials, as well as salt rejection. Suchbehavior opens new exciting leads, which remain to be explored,including e.g., the manipulation of dripplons and embeddedsolutes via applied electric fields or osmotic gradients.

We hope that our results will seed experimental investigationsto search for these predicted exotic structures and their con-sequences on membrane transport.

Data availability. The data that support the findings of this studyare available from the corresponding author on request.

Received: 29 November 2017 Accepted: 15 March 2018

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AcknowledgementsL.B. and B.R. acknowledge funding from the Agence Nationale de la Recherche (ANR),project “Neptune”. This work was granted access to the HPC resources of MesoPSLfinanced by the Region Ile de France and the project Equip@Meso (reference ANR-10-EQPX-29-01) of the program Investissements d’Avenir supervised by ANR.

Author contributionsL.B. conceived the project. H.Y. performed the simulations and H.Y., V.K., B.R., and L.B.analyzed the data. H.Y., B.R., and L.B. wrote the manuscript.

Additional informationSupplementary Information accompanies this paper at https://doi.org/10.1038/s41467-018-03829-1.

Competing interests: The authors declare no competing interests.

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Dripplons as localized and superfast ripples of water confined between graphene sheetsResultsWater confined between rigid graphene sheetsGraphene flexibility and localized dripplonsOstwald ripening and dripplon osmotic stabilizationThermodynamics of dripplon creationSuperdiffusive dripplon dynamics

DiscussionData availability

ReferencesAcknowledgementsAuthor contributionsCompeting interestsACKNOWLEDGEMENTS

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